Fluid limit for the Poisson encounter-mating model
Abstract
Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article, we study SEM with Poisson firing times. First, we prove that the model enjoys a fluid limit as the population size diverges, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Then, we convert these ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Next, under the so-called fine balance condition which characterizes panmixia, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under certain symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to characterize assortative mating.
Cite
@article{arxiv.1411.7220,
title = {Fluid limit for the Poisson encounter-mating model},
author = {Onur Gün and Atilla Yilmaz},
journal= {arXiv preprint arXiv:1411.7220},
year = {2016}
}
Comments
23 pages, 1 figure. We resolved a technical issue in the beginning of Section 2.1, included a self-contained proof of Theorem 2.1, removed the part about the diffusion limit which did not fit the rest of the paper (hence the revised title), corrected/completed the proof of Theorem 4.1, and made various other (minor) changes